In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of mathematical singularities. The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology.
Read more about Analytic Continuation: Initial Discussion, Applications, Formal Definition of A Germ, The Topology of The Set of Germs, Examples of Analytic Continuation, Monodromy Theorem, Hadamard's Gap Theorem, Polya's Theorem
Famous quotes containing the words analytic and/or continuation:
“You, that have not lived in thought but deed,
Can have the purity of a natural force,
But I, whose virtues are the definitions
Of the analytic mind, can neither close
The eye of the mind nor keep my tongue from speech.”
—William Butler Yeats (18651939)
“I believe it was a good job,
Despite this possible horror: that they might prefer the
Preservation of their law in all its sick dignity and their knives
To the continuation of their creed
And their lives.”
—Gwendolyn Brooks (b. 1917)